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Solving Constrained Combinatorial Optimization Problems with Variational Quantum Imaginary Time Evolution

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Pith's one-line read Variational quantum imaginary time evolution solves constrained problems like the Multiple Knapsack Problem with lower optimality gaps than QAOA and VQE.

desk verdict Plausible but under-specified empirical case for VarQITE on constrained optimization; the headline gap numbers are likely biased by an ambiguous infeasible-solution convention and an undisclosed rescaling factor. read the letter →

arxiv 2504.12607 v2 pith:GY2CLQAZ submitted 2025-04-17 quant-ph

Xin Wei Lee , Hoong Chuin Lau This is my paper · ORCID
classification quant-ph
keywords variationalquantumimaginarytimeevolutionVarQITEmultipleknapsackproblemQUBOMax-CutHamiltonianansatzconstrainedcombinatorialoptimizationrescaling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that imaginary time evolution, implemented variationally, is a better optimizer for constrained combinatorial problems than conventional variational quantum eigensolver approaches. It tests this on the Multiple Knapsack Problem with up to three knapsacks and four items, converting each instance into an unbalanced QUBO and then a Max-Cut graph to use the imaginary Hamiltonian variational ansatz. Across 68 instances, the QITE+iHVA protocol achieves feasibility rates comparable to VQE methods but with drastically lower mean optimality gaps. Rescaling the Hamiltonian coefficients further narrows the gap and reduces the number of integration steps needed.

What carries the argument

The framework chains three components: the unbalanced penalization QUBO formulation that avoids slack variables, a conversion of the QUBO into a Max-Cut instance by adding one vertex, and the iHVA ansatz whose RZY gates are assembled from breadth-first spanning trees of the Max-Cut graph. Parameter updates come from the McLachlan variational principle, which solves the linear system M θ̇ = V at each imaginary-time Euler step. Scaling the Hamiltonian by 1/d is shown to be equivalent to taking larger time steps, reducing the number of linear-system solves.

What would settle it

Run the same QITE+iHVA versus VQE comparison on larger MKP instances (e.g., 20 items) and check whether the mean optimality gap advantage persists; or deliberately pick Max-Cut graphs where the spanning-tree-based iHVA is known to be inexpressive and see whether VarQITE's energy stops following imaginary-time evolution.

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Extended reading notes

Core claim

The paper's central claim is that the variational quantum imaginary time evolution (VarQITE) protocol, combined with the Max-Cut-tailored imaginary Hamiltonian variational ansatz (iHVA), finds solutions far closer to the exact optimum than conventional VQE methods. On the 68 tested MKP instances, the median optimality gap for the rescaled QITE+iHVA is roughly 0.2, while HEA and ma-QAOA sit around 2.0-2.5 and iHVA optimized by BFGS near 4.0. The authors attribute this to the McLachlan variational principle, which updates parameters by tracing imaginary-time evolution rather than minimizing a nonconvex expectation landscape.

Load-bearing premise

The iHVA ansatz must be expressive enough to approximate the true imaginary-time-evolved state at every Euler step; if the ansatz manifold is too small, the reported gap advantage could be an artifact of the 10 to 13 qubit instances selected.

Editorial extensions

If this is right

  • Constrained problems expressible as QUBO can be attacked with VarQITE by converting them to Max-Cut, avoiding the ancilla qubits and multi-controlled Toffoli gates of QAOA+.
  • Hamiltonian rescaling acts as a cost-accuracy hyperparameter: moderate scaling (e.g., d = 10) improves solution quality, while larger scaling cuts the number of time steps but can miss the optimum.
  • Because QITE's parameter path bypasses nonconvex gradient landscapes, the method may be less sensitive to random initialization than VQE.
  • The same pipeline could be applied to other inequality-constrained problems such as bin packing or vehicle routing.
  • On real hardware, the bottleneck shifts to estimation of the matrix M and vector V, which require circuit evaluations and classical linear algebra at every step.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the iHVA expressivity assumption holds at scale, the advantage over VQE could grow on larger instances where classical local optimizers get trapped in more local minima.
  • The rescaling trick suggests an adaptive schedule that adjusts d along the evolution could further reduce the number of steps without losing accuracy.
  • The QUBO-to-Max-Cut conversion may be a key enabler: because iHVA exploits Max-Cut's bit-flip symmetry, a direct QUBO ansatz might not reap the same QITE benefit.
  • The claimed advantage is measured on 10-13 qubit instances; a natural test is to see whether the mean optimality gap separation persists on 20+ qubit problems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper proposes a variational quantum imaginary time evolution (VarQITE) framework for the Multiple Knapsack Problem (MKP). The MKP is converted to an unbalanced QUBO and then to a Max-Cut instance so that a Max-Cut-tailored imaginary Hamiltonian variational ansatz (iHVA) can be used. The authors compare QITE+iHVA with iHVA, ma-QAOA, and HEA optimized by VQE on 68 small MKP instances (10-13 qubits), reporting feasibility rates, optimality rates, and mean optimality gaps. They find that QITE+iHVA, especially with rescaled Hamiltonian coefficients, achieves lower mean optimality gaps, and they discuss how scaling the Hamiltonian by 1/d allows larger effective time steps and fewer QITE iterations.

Significance. The paper addresses a timely problem: applying VarQITE to constrained combinatorial optimization, specifically MKP. If the reported results hold, the work would provide one of the first demonstrations that variational imaginary-time evolution can beat conventional VQE/QAOA variants on constrained QUBOs, and the Hamiltonian-rescaling observation in Eq. (22) is a useful practical idea for reducing QITE simulation cost. The benchmark is externally anchored to Gurobi optima, several ansatze are compared under a common workflow, and the rescaling algebra is internally consistent. However, the numerical support is currently weakened by the undefined treatment of infeasible trials in the mean optimality gap and by the undisclosed scale factor d; the experiments are also limited to 10-13 qubits. The iHVA expressivity concern raised by the stress-test note is plausible but secondary on these small instances. With the main reproducibility issues resolved, the qualitative conclusion that QITE+iHVA is competitive with or better than VQE baselines on small MKP instances would be credible.

major comments (3)
  1. [Section III, Definition 3, Eq. (18)] The mean optimality gap is not well-defined for infeasible solutions. The definition substitutes the sampled bit-string back into the original constrained problem; an infeasible string can produce an objective value larger than C_opt, making the gap negative. The text says the average is taken over all trials, while Fig. 3(c) starts its y-axis at 0, so it is not clear whether infeasible trials are excluded, clipped, or counted at face value. Since Table I and Fig. 3(c) are the primary evidence for the central claim, the authors must state the convention explicitly and recompute the mean gaps under a fair convention, such as feasible trials only or mapping infeasible trials to a fixed gap.
  2. [Section IV, Eq. (21), Table I, Fig. 3] The scale factor d used for the 'rescaled QITE+iHVA' results is not reported anywhere in the manuscript. Section IV only says d=10 is best for solution quality and d=100 is best for cost on the six instances in Fig. 4, and the main experiments in Table I and Fig. 3 do not state which d was used or whether it was chosen per instance. Because d is a tunable hyperparameter, the improved mean gap of the rescaled method in Table I is not reproducible and may reflect tuning on the benchmark instances. Please disclose d for the main experiments and provide a sensitivity analysis over d.
  3. [Section II.C, Eq. (17) and Section IV, Eq. (19)] The main QITE experiments do not report the total evolution time tau and the number of time steps N_tau. The experimental settings in Section III list the optimizer, ansatz, and penalty coefficients, but omit these two parameters, although Eq. (19) identifies them as controlling accuracy and cost, and Fig. 4 fixes tau=10. Without these values, the QITE results in Table I cannot be reproduced, and the comparison with VQE methods is under-specified. Please add these settings to Section III and state whether they were fixed across instances.
minor comments (4)
  1. [Section II.C, Section III] The expressivity assumption of iHVA is stated but not empirically checked; a sentence acknowledging that iHVA may not track the imaginary-time-evolved state for larger instances would help scope the claims.
  2. [Section III, Fig. 3(c) and Table I] The text says 'the mean optimality gap of QITE is drastically lower than those of the classical VQE methods,' but the table reports 0.64 for QITE+iHVA and 0.31 for rescaled QITE+iHVA; please clarify which version is meant in each statement.
  3. [Table I caption] The column headers 'Feasibility' and 'Optimality' would be clearer if they repeated 'best out of 5' in both the header row and the caption, since the table uses two different notions of feasibility/optimality rates.
  4. [General] There is no data or code availability statement; providing the 68 MKP instances and the implementation would significantly improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central benchmark is anchored to externally computed Gurobi optima, and the QITE parameter-update rule is an independent variational principle not fitted to the target results.

full rationale

The paper's central claim is an empirical comparison against Gurobi-optimal solutions (Definition 3 and Table I), so the reported optimality gaps are anchored to an external classical solver rather than derived from the method's own assumptions. The VarQITE update rule (Eqs. 12-17) is the standard McLachlan variational principle applied to the problem Hamiltonian; it does not assume the target optimality-gap conclusion. The iHVA ansatz is adopted from external prior work by Wang et al. [31], not from the present authors, and the QUBO-to-Max-Cut conversion is taken from Barahona et al. [39]. The few self-citations ([17], [18], [21], [23]) appear only in contextual lists of parameter-initialization and ansatz heuristics; none of them supplies a load-bearing premise for the QITE advantage. The Hamiltonian rescaling in Eq. (21) is a hyperparameter d whose value is chosen empirically, and the paper explicitly acknowledges this limitation in the conclusion ('heuristics that cleverly determine the scale of the Hamiltonian are required, instead of finding the scales empirically'); empirical tuning is a reproducibility concern, not circularity. The possible issue that infeasible samples could enter the mean optimality gap of Eq. (18) is an evaluation-metric concern, not a case of a prediction reducing to its inputs. No quoted equation or cited result makes the derivation equivalent to its own conclusion, so no circular step can be exhibited and the appropriate score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a chain of modeling choices: the unbalanced QUBO penalty must encode feasibility correctly, the QUBO-to-Max-Cut conversion must preserve the optimum, the iHVA manifold must contain the relevant imaginary-time-evolved states, and the chosen integration parameters must be accurate. The paper does not prove the first, acknowledges the third as the core challenge, and leaves d and Ntau under-specified.

free parameters (4)
  • Penalty coefficients lambda1 and lambda2 in unbalanced QUBO = lambda1 = lambda2 = 10
    Set by hand for all instances, not derived. Controls the trade-off between objective and constraint satisfaction; the central performance comparison uses this fixed choice.
  • Hamiltonian scale factor d for rescaled QITE+iHVA = not reported for Table I; grid {1, 10, 100, 1000} explored in Figure 4
    The rescaled variant's performance depends on d. Figure 4 selects d=10 for best solution quality and d=100 for best cost, but the benchmark does not disclose the d used for the headline results.
  • Total imaginary time tau and time-step count Ntau = tau=10; Ntau varied up to 500 in Figure 4, not stated for Figure 3
    QITE integration hyperparameters chosen manually. The number of linear-system solves per instance is a core determinant of optimization cost, and it is not pinned down for the 68-instance comparison.
  • Ansatz repetition depth p = p=1
    Set to one repetition for all methods; depth affects expressivity and the central comparison.
assumptions (4)
  • domain assumption The unbalanced penalization with Taylor-expanded exponential e^{-h} approx 1 - lambda1 h + lambda2 h^2 and lambda1=lambda2=10 produces a QUBO whose ground states correspond to the original MKP optima.
    Invoked in Equation (6). No feasibility-guarantee proof is given for lambda=10; the approximation is nonlinear and the penalty may distort the solution landscape.
  • standard math A QUBO instance with n variables can be mapped losslessly to a Max-Cut instance with n+1 vertices.
    Taken from Barahona et al. and used in Section II.A and Algorithm 1. The transformation is exact if the coefficients are handled correctly.
  • domain assumption The iHVA ansatz is expressive enough to represent the imaginary-time-evolved quantum state of the MKP-derived Max-Cut Hamiltonian at every integration step.
    Section II.B constructs iHVA from the Max-Cut graph's spanning trees; Section II.C assumes an expressive unitary. The paper itself notes the expressivity challenge in Section III.
  • domain assumption Euler integration of the McLachlan equations with step Delta tau is an accurate surrogate for exact imaginary time evolution for all instances tested.
    Section II.C Equation (17). Accuracy depends on Delta tau and the spectral norm; the paper relies on this to interpret converged parameters as ground states.

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Pith. "Pith review of Solving Constrained Combinatorial Optimization Problems with Variational Quantum Imaginary Time Evolution." pith.science (2026). https://pith.science/paper/GY2CLQAZ

@misc{pith2026250412607,
  author       = {Pith},
  title        = {Pith review of: Solving Constrained Combinatorial Optimization Problems with Variational Quantum Imaginary Time Evolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GY2CLQAZ}},
  note         = {Machine review of arXiv:2504.12607}
}
read the original abstract

Solving combinatorial optimization problems using variational quantum algorithms (VQAs) has emerged as a promising research direction. Since the introduction of the Quantum Approximate Optimization Algorithm (QAOA), numerous variants have been proposed to enhance its performance. QAOA was later extended to the Quantum Alternating Operator Ansatz (QAOA+), which generalizes the initial state, phase-separation operator, and mixer to address constrained problems without relying on the standard Quadratic Unconstrained Binary Optimization (QUBO) formulation. However, QAOA+ often requires additional ancilla qubits and a large number of multi-controlled Toffoli gates to prepare the superposition of feasible states, resulting in deep circuits that are challenging for near-term quantum devices. Furthermore, VQAs are generally hindered by issues such as barren plateaus and suboptimal local minima. Recently, Quantum Imaginary Time Evolution (QITE), a ground-state preparation algorithm, has been explored as an alternative to QAOA and its variants. QITE has demonstrated improved performance in quantum chemistry problems and has been applied to unconstrained combinatorial problems such as Max-Cut. In this work, we apply the variational form of QITE (VarQITE) to solve the Multiple Knapsack Problem (MKP), a constrained problem, using a Max-Cut-tailored ansatz. To the best of our knowledge, this is the first attempt to address constrained optimization using VarQITE. We show that VarQITE achieves significantly lower mean optimality gaps compared to QAOA and other conventional methods. Moreover, we demonstrate that scaling the Hamiltonian coefficients can further reduce optimization costs and accelerate convergence.

Figures

Figures reproduced from arXiv: 2504.12607 by the authors.

Figure 1
Figure 1. Workflow for the framework used in this work. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The convergence of QITE+iHVA to the solution [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Performance comparison of different methods (HEA, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Effect of scaling the Hamiltonian coefficients by [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.